The Basis Number of the Cartesian Product of Different Ladders

Abstract

The basis number of a graph \(G\) is defined to be the least positive integer \(d\) such that \(G\) has a \(d\)-fold basis for the cycle space of \(G\).

In this paper, we prove that the basis number of the Cartesian product of different ladders is exactly \(4\). However, if we apply Theorem \(4.1\) of Ali and Marougi \([4]\), which is stated in the introduction as Theorem \(1.1\), we find that the basis number of the circular and Möbius ladders with circular ladders and Möbius ladders is less than or equal to \(5\), and the basis number of ladders with circular ladders and circular ladders with circular ladders is at most \(4\).